Circles - GRE Quantitative Reasoning

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Question

What is the circumference of a circle with an area of 36π?

Answer

We know that the area of a circle can be expressed: a = πr2

If we know that the area is 36π, we can substitute this into said equation and get: 36π = πr2

Solving for r, we get: 36 = r2; (after taking the square root of both sides:) 6 = r

Now, we know that the circuference of a circle is expressed: c = πd. Since we know that d = 2r (two radii, placed one after the other, make a diameter), we can rewrite the circumference equation to be: c = 2πr

Since we have r, we can rewrite this to be: c = 2π*6 = 12π

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Question

What is the area of a circle, one-quarter of the circumference of which is 5.5 inches?

Answer

Here, you need to “solve backward” from the data you have been given. We know that 0.25C = 5.5; therefore, C = 22. In order to solve for the area, we will need the radius of the circle. This can be obtained by recalling that C = 2πr. Replacing 22 for C, we get 22 = 2πr.

Solve for r: r = 22 / 2π = 11 / π.

Now, we solve for the area: A = πr2. Replacing 11 / π for r: A = π (11 / π)2 = (121π) / (π2) = 121 / π.

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Question

Quantitative Comparison

Quantity A: Area of a right triangle with sides 7, 24, 25

Quantity B: Area of a circle with radius 5

Answer

Quantity A: area = base * height / 2 = 7 * 24/2 = 84

Quantity B: area = πr_2 = 25_π

Now we have to remember what π is. Using π = 3, the area is approximately 75. Using π = 3.14, the area increases a little bit, but no matter how exact an approximation for π, this area will never be larger than Quantity A.

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Question

If the outer arc of 1/12th of a circular pie is 7π, what is the area of 1/4th of the pie?

Answer

Our initial data tells us that (1/12)c = 7π or (1/12)πd = 7π. This simplifies to (1/12)d = 7 or d = 84. Furthermore, we know that r is 42. Given this, we can ascertain the area of a quarter of the whole pie by taking one fourth of the whole area or 0.25 * π * 422 = 0.25 * 1764 * π = 441π

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Question

Circle A has an area of $121\pi$ . What is the perimeter of an enclosed semi-circle with half the radius of circle A?

Answer

Based on our information, we know that the 121π = πr2; 121 = r2; r = 11.

Our other circle with half the radius of A has a diameter equal to the radius of A. Therefore, the circumference of this circle is 11π. Half of this is 5.5π. However, since this is a semi circle, it is enclosed and looks like this:

Therefore, we have to include the diameter in the perimeter. Therefore, the total perimeter of the semi-circle is 5.5π + 11.

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Question

Which is greater: the circumference of a circle with an area of $25\pi \ in^2$ , or the perimeter of a square with side length inches?

Answer

Starting with the circle, we need to find the radius in order to get the circumference. Find $\dpi{100} \small r$ by plugging our given area into the equation for the area of a circle:

$A = \pi r^2$

$25\pi = \pi r^2$

$25 = r^2$

$r = 5\ in$

Then calculate circumference:

$C = 2\pi r$

$\dpi{100} \small C = 2\pi \times 5 = 10\pi \approx 31.4 inches$ (approximating $\dpi{100} \small \pi$ as 3.14)

To find the perimeter of the square, we can use $P = 4s$ , where $\dpi{100} \small P$ is the perimeter and $\dpi{100} \small s$ is the side length:

$P = 4 \times 7\ in = 28\ in$

$\dpi{100} \small 31.4>28$ , so the circle's circumference is greater.

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Question

$x\ge 1$

Quantity A: The circumference of a circle with radius

Quantity B: The area of a circle with a diameter one fourth the radius of the circle in Quantity A

Which of the following is true?

Answer

Let's compute each value separately. We know that the radii are positive numbers that are greater than or equal to . This means that we do not need to worry about the fact that the area could represent a square of a decimal value like .

Quantity A

Since $c=2\pi r$ , we know:

$a=2 \cdot \pi \cdot 24x = 48x\pi$

Quantity B

If the diameter is one-fourth the radius of A, we know:
$d = \frac{1}{4}\cdot 24x = 6x$

Thus, the radius must be half of that, or .

Now, we need to compute the area of this circle. We know:

$A = \pi r^2$

Therefore, $A = \pi \cdot (3x)^2 = 9x^2$

Now, notice that if , Quantity A is larger.

However, if we choose a value like , we have:

Quantity A: $48 * 1000 \pi = 48000\pi$

Quantity B: $9 * 1000^2\pi= 9000000\pi$

Therefore, the relation cannot be determined!

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Question

Circle has a center in the center of Square .

The area of Square is .

What is the circumference of Circle ?

Answer

Since we know that the area of Square is , we know , where is the length of one of its sides. From this, we can solve for by taking the square root of both sides. You will have to do this by estimating upward. Therefore, you know that is . By careful guessing, you can quickly see that is . From this, you know that the diameter of your circle must be half of , or (because it is circumscribed). Therefore, you can draw:

The circumference of this circle is defined as:

$C=2\pi r$ or, for your values:

$c=2\pi * 12 = 24\pi$

(You could also compute this from the diameter, but many students just memorize the formula above.)

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Question

Circle A has an area of $121\pi$ . What is the perimeter of an enclosed semi-circle with half the radius of circle A?

Answer

Based on our information, we know that the 121π = πr2; 121 = r2; r = 11.

Our other circle with half the radius of A has a diameter equal to the radius of A. Therefore, the circumference of this circle is 11π. Half of this is 5.5π. However, since this is a semi circle, it is enclosed and looks like this:

Therefore, we have to include the diameter in the perimeter. Therefore, the total perimeter of the semi-circle is 5.5π + 11.

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Question

Which is greater: the circumference of a circle with an area of $25\pi \ in^2$ , or the perimeter of a square with side length inches?

Answer

Starting with the circle, we need to find the radius in order to get the circumference. Find $\dpi{100} \small r$ by plugging our given area into the equation for the area of a circle:

$A = \pi r^2$

$25\pi = \pi r^2$

$25 = r^2$

$r = 5\ in$

Then calculate circumference:

$C = 2\pi r$

$\dpi{100} \small C = 2\pi \times 5 = 10\pi \approx 31.4 inches$ (approximating $\dpi{100} \small \pi$ as 3.14)

To find the perimeter of the square, we can use $P = 4s$ , where $\dpi{100} \small P$ is the perimeter and $\dpi{100} \small s$ is the side length:

$P = 4 \times 7\ in = 28\ in$

$\dpi{100} \small 31.4>28$ , so the circle's circumference is greater.

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Question

$x\ge 1$

Quantity A: The circumference of a circle with radius

Quantity B: The area of a circle with a diameter one fourth the radius of the circle in Quantity A

Which of the following is true?

Answer

Let's compute each value separately. We know that the radii are positive numbers that are greater than or equal to . This means that we do not need to worry about the fact that the area could represent a square of a decimal value like .

Quantity A

Since $c=2\pi r$ , we know:

$a=2 \cdot \pi \cdot 24x = 48x\pi$

Quantity B

If the diameter is one-fourth the radius of A, we know:
$d = \frac{1}{4}\cdot 24x = 6x$

Thus, the radius must be half of that, or .

Now, we need to compute the area of this circle. We know:

$A = \pi r^2$

Therefore, $A = \pi \cdot (3x)^2 = 9x^2$

Now, notice that if , Quantity A is larger.

However, if we choose a value like , we have:

Quantity A: $48 * 1000 \pi = 48000\pi$

Quantity B: $9 * 1000^2\pi= 9000000\pi$

Therefore, the relation cannot be determined!

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Question

Circle has a center in the center of Square .

The area of Square is .

What is the circumference of Circle ?

Answer

Since we know that the area of Square is , we know , where is the length of one of its sides. From this, we can solve for by taking the square root of both sides. You will have to do this by estimating upward. Therefore, you know that is . By careful guessing, you can quickly see that is . From this, you know that the diameter of your circle must be half of , or (because it is circumscribed). Therefore, you can draw:

The circumference of this circle is defined as:

$C=2\pi r$ or, for your values:

$c=2\pi * 12 = 24\pi$

(You could also compute this from the diameter, but many students just memorize the formula above.)

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Question

Circle A has an area of $121\pi$ . What is the perimeter of an enclosed semi-circle with half the radius of circle A?

Answer

Based on our information, we know that the 121π = πr2; 121 = r2; r = 11.

Our other circle with half the radius of A has a diameter equal to the radius of A. Therefore, the circumference of this circle is 11π. Half of this is 5.5π. However, since this is a semi circle, it is enclosed and looks like this:

Therefore, we have to include the diameter in the perimeter. Therefore, the total perimeter of the semi-circle is 5.5π + 11.

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Question

Which is greater: the circumference of a circle with an area of $25\pi \ in^2$ , or the perimeter of a square with side length inches?

Answer

Starting with the circle, we need to find the radius in order to get the circumference. Find $\dpi{100} \small r$ by plugging our given area into the equation for the area of a circle:

$A = \pi r^2$

$25\pi = \pi r^2$

$25 = r^2$

$r = 5\ in$

Then calculate circumference:

$C = 2\pi r$

$\dpi{100} \small C = 2\pi \times 5 = 10\pi \approx 31.4 inches$ (approximating $\dpi{100} \small \pi$ as 3.14)

To find the perimeter of the square, we can use $P = 4s$ , where $\dpi{100} \small P$ is the perimeter and $\dpi{100} \small s$ is the side length:

$P = 4 \times 7\ in = 28\ in$

$\dpi{100} \small 31.4>28$ , so the circle's circumference is greater.

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Question

$x\ge 1$

Quantity A: The circumference of a circle with radius

Quantity B: The area of a circle with a diameter one fourth the radius of the circle in Quantity A

Which of the following is true?

Answer

Let's compute each value separately. We know that the radii are positive numbers that are greater than or equal to . This means that we do not need to worry about the fact that the area could represent a square of a decimal value like .

Quantity A

Since $c=2\pi r$ , we know:

$a=2 \cdot \pi \cdot 24x = 48x\pi$

Quantity B

If the diameter is one-fourth the radius of A, we know:
$d = \frac{1}{4}\cdot 24x = 6x$

Thus, the radius must be half of that, or .

Now, we need to compute the area of this circle. We know:

$A = \pi r^2$

Therefore, $A = \pi \cdot (3x)^2 = 9x^2$

Now, notice that if , Quantity A is larger.

However, if we choose a value like , we have:

Quantity A: $48 * 1000 \pi = 48000\pi$

Quantity B: $9 * 1000^2\pi= 9000000\pi$

Therefore, the relation cannot be determined!

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Question

Circle has a center in the center of Square .

The area of Square is .

What is the circumference of Circle ?

Answer

Since we know that the area of Square is , we know , where is the length of one of its sides. From this, we can solve for by taking the square root of both sides. You will have to do this by estimating upward. Therefore, you know that is . By careful guessing, you can quickly see that is . From this, you know that the diameter of your circle must be half of , or (because it is circumscribed). Therefore, you can draw:

The circumference of this circle is defined as:

$C=2\pi r$ or, for your values:

$c=2\pi * 12 = 24\pi$

(You could also compute this from the diameter, but many students just memorize the formula above.)

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Question

An ant begins at the center of a pie with a 12" radius. Walking out to the edge of pie, it then proceeds along the outer edge for a certain distance. At a certain point, it turns back toward the center of the pie and returns to the center point. Its whole trek was 55.3 inches. What is the approximate size of the angle through which it traveled?

Answer

To solve this, we must ascertain the following:

The arc length through which the ant traveled.

The percentage of the total circumference in light of that arc length.

The percentage of 360° proportionate to that arc percentage.

To begin, let's note that the ant travelled 12 + 12 + x inches, where x is the outer arc distance. (It traveled the radius twice, remember); therefore, we know that 24 + x = 55.3 or x = 31.3.

Now, the total circumference of the circle is 2πr or 24π. The arc is 31.3/24π percent of the total circumference; therefore, the percentage of the angle is 360 * 31.3/24π. Since the answers are approximations, use 3.14 for π. This would be 149.52°.

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Question

A study was conducted to determine the effectiveness of a vaccine for the common cold (Rhinovirus sp.). 1000 patients were studied. Of those, 500 received the vaccine and 500 did not. The patients were then exposed to the Rhinovirus and the results were tabulated.

Table 1 shows the number of vaccinated and unvaccinated patients in each age group who caught the cold.

Suppose the scientists wish to create a pie chart reflecting a patient's odds of catching the virus depending on vaccination status and age group.

All 1000 patients are included in this pie chart.

What would be the angle of the arc for the portion of the chart representing vaccinated patients of all age groups who caught the virus?

Answer

First, we must determine what proportion of the 1000 patients were vaccinated and caught the virus. The total number of patients who were vaccinated and caught the virus is 50.

18 + 4 + 5 + 4 + 19 = 50

The proportion of the patients is represented by dividing this group by the total number of participants in the study.

50/1000 = 0.05

Next, we need to figure out how that proportion translates into a proportion of a pie chart. There are 360° in a pie chart. Multiply 360° by our proportion to reach the solution.

360° * 0.05 = 18°

The angle of the arc representing vaccinated patients who caught the virus is 18°.

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Question

A group of students ate an -inch pizza that was cut into equal slices. What was the angle measure needed to cut this pizza into these equal slices?

Answer

You will not need all of the information given in the prompt in order to answer this question successfully. You really only need to know that there were slices. If the slices were evenly divided among the degrees of the pizza, this means that the degree measure of each slice was $\frac{360}{14}$ . This reduces to $\frac{180}{7}$ degrees.

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Question

An ant begins at the center of a pie with a 12" radius. Walking out to the edge of pie, it then proceeds along the outer edge for a certain distance. At a certain point, it turns back toward the center of the pie and returns to the center point. Its whole trek was 55.3 inches. What is the approximate size of the angle through which it traveled?

Answer

To solve this, we must ascertain the following:

The arc length through which the ant traveled.

The percentage of the total circumference in light of that arc length.

The percentage of 360° proportionate to that arc percentage.

To begin, let's note that the ant travelled 12 + 12 + x inches, where x is the outer arc distance. (It traveled the radius twice, remember); therefore, we know that 24 + x = 55.3 or x = 31.3.

Now, the total circumference of the circle is 2πr or 24π. The arc is 31.3/24π percent of the total circumference; therefore, the percentage of the angle is 360 * 31.3/24π. Since the answers are approximations, use 3.14 for π. This would be 149.52°.

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